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Adaptive output-feedback stabilization of non-local hyperbolic PDEs

AUTOMATICA 2014年第10期50卷 2692-2699页

作者： Bernard, Pauline Krstic, Miroslav MINES ParisTech F-75006 Paris France Univ Calif San Diego Dept Mech & Aerosp Engn La Jolla CA 92093 USA

We address the problem of adaptive output-feedback stabilization of general first-order hyperbolic partial integro-differential equations (PIDE). Such systems are also referred to as PDEs with non-local (in space) terms. We apply control at one boundary, take measurements on the other boundary, and allow the system's functional coefficients to be unknown. To deal with the absence of both full-state measurement and parameter knowledge, we introduce a pre-transformation (which happens to be based on backstepping) of the system into an observer canonical form. In that form, the problem of adaptive observer design becomes tractable. Both the parameter estimator and the control law employ only the input and output signals (and their histories over one unit of time). Prior to presenting the adaptive design, we present the non-adaptive/baseline controller, which is novel in its own right and facilitates the understanding of the more complex, adaptive system. The parameter estimator is of the gradient type, based on a parametric model in the form of an integral equation relating delayed values of the input and output. For the closed-loop system we establish boundedness of all signals, pointwise in space and time, and convergence of the POE state to zero pointwise in space. We illustrate our result with a simulation. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： distributed parameter systems Boundary control Hyperbolic systems Adaptive control

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parameter identification of aggregated thermostatically controlled loads for smart grids using PDE techniques

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INTERNATIONAL JOURNAL OF CONTROL 2014年第7期87卷 1373-1386页

作者： Moura, Scott Bendtsen, Jan Ruiz, Victor Univ Calif Berkeley Dept Civil & Environm Engn Berkeley CA 94720 USA Aalborg Univ Dept Elect Syst Automat & Control DK-9220 Aalborg Denmark Univ Calif San Diego Dept Mech & Aerosp Engn La Jolla CA 92093 USA

This paper develops methods for model identification of aggregated thermostatically controlled loads (TCLs) in smart grids, via partial differential equation (PDE) techniques. Control of aggregated TCLs provides a promising opportunity to mitigate the mismatch between power generation and demand, thus enhancing grid reliability and enabling renewable energy penetration. To this end, this paper focuses on developing parameter identification algorithms for a PDE-based model of aggregated TCLs. First, a two-state boundary-coupled hyperbolic PDE model for homogenous TCL populations is derived. This model is extended to heterogeneous populations by including a diffusive term, which provides an elegant control-oriented model. Next, a passive parameter identification scheme and a swapping-based identification scheme are derived for the PDE model structure. Simulation results demonstrate the efficacy of each method under various autonomous and non-autonomous scenarios. The proposed models can subsequently be employed to provide system critical information for power system monitoring and control.

关键词： distributed parameter systems demand side management model identification smart grids parameter estimation

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Evolution of regulation errors in the diffusion layer in electrodeposition process control

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JOURNAL OF PROCESS CONTROL 2014年第7期24卷 1098-1105页

作者： Tenno, R. Aalto Univ Sch Elect Engn Aalto Finland

A random field induced by infinitely many noises emerges in numerous processes, such as a mixed boundary electrochemical deposition process, which is determined by the main reaction and several side reactions. Since the process is controlled on one boundary without knowledge of the side reactions, a model error emerges and the regulation error trails. The model considered in this paper allows for general settings and is defined for white noise and colour noise. The white noise is responsible for unpredictable model errors and the colour noise for systematic errors. A relatively simple boundary control is applied that, when lifted into the domain, acts as a smooth function. The paper demonstrates that the regulation error can be suppressed effectively even if the correlation of random noises is relatively weak. The regulation error is suppressed by the double effects of the Laplacian and noise correlation. The traces of covariance operators are found for several categories of the domain noises and for the boundary noises. A similar trace function in general settings satisfies a Kolmogorov equation in infinitely many state variables. The solution for the Kolmogorov equation is simplified to a form that makes numerical treatment possible. The regulation errors induced by white noise are compared with the errors induced by coloured noise in the numerical analysis. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Stochastic process distributed parameter systems Process control Electrodeposition

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Bending moment-based force control of flexible arm under gravity

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MECHANISM AND MACHINE THEORY 2014年 79卷 217-229页

作者： Endo, Takahiro Kawasaki, Haruhisa Gifu Univ Fac Engn Gifu 5011193 Japan

We discuss the force control problem for a constrained one-link flexible arm with the effect of gravity, based on a distributed parameter model. To solve the force control problem, we propose a simple controller constructed using only the bending moment at the root of the flexible arm. Information about the force and rotational angle of the motor is not necessary for implementation of the controller, and thus a force sensor and encoder are not needed. Several numerical simulations were carried out to determine the performance of the proposed controller, and we found that it works well for force control. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Force control Flexible arm distributed parameter systems

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Non-quadratic exponential stabilisation of non-linear hyperbolic partial differential equation systems

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IET SCIENCE MEASUREMENT & TECHNOLOGY 2014年第6期8卷 537-545页

作者： Sadeghi, Mokhtar Sha Vafamand, Navid Babaei, Mohammad Sadegh Shiraz Univ Technol Elect & Elect Dept Shiraz Iran Shiraz Univ E Learning Dept Shiraz Iran

In this study, a new systematic approach is proposed to design the fuzzy controller for a class of Takagi-Sugeno fuzzy-partial differential equation (TS fuzzy-PDE) systems which describe the non-linear distributed parameter system formulated by first-order semi-linear hyperbolic PDEs. In this study, non-quadratic Lyapunov function is utilised and some slack matrices are introduced to derive stability conditions in terms of linear matrix inequalities (LMIs). The proposed approach has three main features. First, stability conditions are not derived in the form of spatial differential LMI. Second, conservativeness of LMI conditions is reduced. Third, there is no restriction on the form of semi-linear hyperbolic PDE systems and therefore more semi-linear systems classes can be stabilised. Also, the proposed approach is more suitable for practical implementation compared with the recently published papers.

关键词： partial differential equations nonlinear differential equations hyperbolic equations fuzzy control control system synthesis nonlinear control systems distributed parameter systems Lyapunov methods linear matrix inequalities asymptotic stability nonquadratic exponential stabilisation nonlinear hyperbolic partial differential equation systems fuzzy controller design systematic approach TakagiaEuro"Sugeno fuzzy partial differential equation TS fuzzy-PDE system nonlinear distributed parameter system first order semilinear hyperbolic PDE nonquadratic Lyapunov function stability conditions linear matrix inequalities LMI

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Linear matrix inequalities-based membership-function-dependent stability analysis for non-parallel distributed compensation fuzzy-model-based control systems

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IET CONTROL THEORY AND APPLICATIONS 2014年第8期8卷 614-625页

作者： Lam, Hak Keung Wu, Ligang Zhao, Yanbin Kings Coll London Dept Informat London WC2R 2LC England Harbin Inst Technol Space Control & Inertial Technol Res Ctr Harbin 150001 Peoples R China

This study investigates the stability of fuzzy-model-based (FMB) control system, which is formed by a Takagi-Sugeno (T-S) fuzzy model representing a non-linear plant and a fuzzy controller connected in a closed loop. The proposed fuzzy controller considers a higher design flexibility that the premise membership functions and the number of fuzzy rules are allowed to be not the same as those of the T-S fuzzy model. By employing simple membership functions and a smaller number of fuzzy rules, the complexity of the fuzzy controller can be reduced to facilitate its implementation. Based on the Lyapunov stability theory, a membership-function-dependent (MFD) stability analysis is proposed to develop stability conditions in the form of linear matrix inequalities (LMIs) to determine the system stability and facilitate the control synthesis. The information of the membership functions is brought to the MFD LMI-based stability conditions, which are dedicated to the FMB control system under consideration. It is thus the proposed MFD stability analysis result is more relaxed compared with the existing membership-function-independent stability analysis results. A simulation example is given to illustrate the effectiveness of the proposed approach.

关键词： linear matrix inequalities stability fuzzy control compensation closed loop systems Lyapunov methods control system synthesis fuzzy set theory distributed parameter systems linear matrix inequalities membership-function-dependent stability analysis nonparallel distributed compensation fuzzy-model-based control systems FMB control system T-S model Takagi-Sugeno model closed loop system MFD LMI-based conditions Lyapunov theory membership-function-independent stability analysis design flexibility fuzzy rules control synthesis

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Optimal dynamic inversion-based boundary control design for two-dimensional heat equations

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IET CONTROL THEORY AND APPLICATIONS 2014年第17期8卷 2013-2025页

作者： Kumar, Manoj Balakrishnan, Sivasubramanya N. Missouri Univ Sci & Technol Dept Mech & Aerosp Engn Rolla MO 65409 USA

This study develops a new boundary control design for a class of distributed parameter systems. A two-dimensional (2D) heat equation is considered and the controller expression is derived for two different types of boundary conditions. The principles of dynamic inversion and optimisation theory are combined to develop an analytical expression for boundary control. To extend the applicability of the controller design to real-life systems, adaptive control theory is used to ensure stability in the presence of parameter uncertainty in the system. L-2 stability is proved using Lyapunov stability theory, which assures that the control drives the system trajectory to the desired temperature profile with uncertainty in the system parameter. Numerical results are presented for a square shape and also a model problem of a 2D cavity.

关键词： distributed parameter systems Lyapunov methods stability adaptive control optimal control optimisation optimal dynamic inversion-based boundary control design two-dimensional heat equations distributed parameter systems 2D heat equation optimisation theory adaptive control theory parameter uncertainty L-2 stability Lyapunov stability theory

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Adaptive Projection-Based Observers and L₁ Adaptive Controllers for Infinite-Dimensional systems With Full-State Measurement

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2014年第3期59卷 585-598页

作者： Natarajan, Vivek Bentsman, Joseph Univ Illinois Urbana IL 61801 USA Univ Illinois Dept Mech Sci & Engn Urbana IL 61801 USA

Adaptive observers using projection-operator-based parameter update laws are considered for a class of linear infinite-dimensional systems with bounded input operator and full state measurement and subject to time-varying matched uncertainties and disturbances. The adaptive control architecture, introduced recently for finite-dimensional plants to provide guaranteed transient performance via fast adaptation, is then extended to this class using the proposed observers. Existence and uniqueness of solutions for the resulting closed loop system and uniform boundedness of the observation error are established first. Then, provided certain assumptions on the plant transfer function and the solution of a Lyapunov inequality hold, uniform guaranteed transient performance bounds on the plant state and control signal under the architecture are derived. Two examples satisfying the assumptions-control of a heat equation and a wave equation-are presented. Reference input tracking simulation results for the heat equation under the adaptive control subject to time-varying matched uncertainties and disturbances are presented in support of the theory.

关键词： Adaptive observers distributed parameter systems guaranteed transient performance L-1 adaptive control projection operator time-varying uncertainty

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Compensation of Wave Actuator Dynamics for Nonlinear systems

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2014年第6期59卷 1555-1570页

作者： Bekiaris-Liberis, Nikolaos Krstic, Miroslav Univ Calif San Diego Dept Mech & Aerosp Engn La Jolla CA 92093 USA

The problem of stabilization of PDE-ODE cascades has been solved in the linear case for several PDE classes, whereas in the nonlinear case the problem has been solved only for the transport/delay PDE, namely for compensation of an arbitrary delay at the input of a nonlinear plant. Motivated by a specific engineering application in off-shore drilling, we solve the problem of stabilization of the cascade of a wave PDE with a general nonlinear ODE. Due to the presence of nonlinearities of arbitrary growth and the time-reversibility of the wave PDE, and due to the possibility of using arguments based on Lyapunov functionals or explicit solutions, several stability analysis approaches are possible. We present stability results in the H-2 x H-1 and C-1 x C-0 norms for general nonlinear ODEs, as well as in the H-1 x L-2 norm for linear ODEs. We specialize our general design for wave PDE-ODE cascades to the case of a wave PDE whose uncontrolled end does not drive an ODE but is instead governed by a nonlinear Robin boundary condition (a "nonlinear spring," as in the friction law in drilling). This is the first global stabilization result for wave equations that incorporate non-collocated destabilizing nonlinearities of superlinear growth. We present two numerical examples, one with a nonlinear ODE and one with a nonlinear spring at the uncontrolled boundary of the wave PDE.

关键词： distributed parameter systems nonlinear control systems delay systems

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Linear matrix inequalities (LMIs) observer and controller design synthesis for parabolic PDE

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EUROPEAN JOURNAL OF CONTROL 2014年第5期20卷 227-236页

作者： Yang, Yu Dubljevic, Stevan MIT Dept Chem Engn Cambridge MA 02139 USA Univ Alberta Dept Chem & Mat Engn Edmonton AB T6G 2V4 Canada

In this work, an observer based controller synthesis is proposed based on the linear matrix inequality (LMI) framework to achieve stabilization of the parabolic PDE in the presence of input constraints. A novel feature of the proposed synthesis is to construct the LMI formulation within the modal space by converting the original parabolic PDE into an infinite-dimensional abstract state space setting. The state feedback controller and the Luenberger observer are developed by accounting for the entire infinite number of modal states, thereby stabilizing the system and reconstructing the state rigorously. The input constraints naturally existing in realistic applications are considered in the design framework. Finally, since initial modal states are hardly known in advance, the LMI formulation, based on the augmented modal state space, including unstable modal states, its estimation error and fast modal output, is developed to maximize the region of attraction (ROA) for the real process state, such that the controller and the observer are robust enough to the error in the initial state guess. By considering a numerical example of an unstable parabolic PDE, we demonstrate that if feasible, the state feedback controller and the observer, generated by the LMI formulation, have capability to drive the process state to the equilibrium steady state. (C) 2014 European Control Association. Published by Elsevier Ltd. All rights reserved.

关键词： Linear matrix inequality Boundary control distributed parameter systems Input constraints Modal state observer

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